Suppose that after submitting a paper for publication, but before hearing anything back from the referee, I discover an error in the paper that needs to be corrected, or an omission that needs to be rectified. What is the best course of action? Should I send a revised version to the editor to forward to the referee, explaining exactly what I have changed? Or wait to hear back from the referee and then include the fixes in my next version along with whatever other changes the referee suggests (again explaining them to the editor)?

I'd really like to hear about this from the point of view of an actual editor. The question here is how much of a pain is it to the editor to have to forward a correction. If the author and referee were in direct contact (say through an online system) I think you'd want to send the correction immediately. The problem is that you don't want to waste the editors time (who typically is a very busy and important person).
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Noah SnyderApr 21 '10 at 18:20

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Speaking as someone who does a job for a journal involving passing papers between authors and referees (a sort of "sub-editor"), I can say that there's few things that annoy me more than authors sending revisions and asking me to pass them on to referees. It confuses things in many ways. For example there is now more than one version of the paper in the system, so when the referee says "the x on page 12 line13 should be y" the author says "no it shouldn't" but it turns out he's looking at another line. And imagine how the referee feels---their job is hard enough without multiple versions!WAIT!
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Kevin BuzzardApr 21 '10 at 18:23

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And a note to the questioner: if you start changing your MS (e.g. updating it on ArXiv) then be sure to keep hold of a copy of the exact version you sent the journal, because that's what all the referee's "page 12 line 13" comments will be based on!
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Kevin BuzzardApr 21 '10 at 18:25

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@Kevin - This is why, instead of page and line number information, the referee should give five to ten words of context with each comment. Typing context/searching for a text string is much, much easier than counting lines of a pdf file (do the equations count?) and is fairly stable across versions. <p> I've started doing this with co-authors and on my referee reports and I'd like all of you to do the same. (Especially if you are my co-author or refereeing one of my papers! :)
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Sam NeadApr 21 '10 at 19:59

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As a referee I always dreamed of cutting out the middleman (i.e. the editor), so that I could communicate with the author directly, while staying anonymous of course. It would save so much refereeing time. In real life I think twice before asking author a question through an editor, because it involves troubling the important people, who may easily get annoyed.
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Igor BelegradekApr 21 '10 at 20:03

6 Answers
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I think Angelo is right in broad strokes. For minor errors it's annoying to bother the editor and the referee. But if it's something that the referee might end up wasting hours on then correcting it pre-emptively makes sense. Here are some examples of things that I'd wait or not on for concreteness. I'd love to hear from people with more knowledge/experience about whether this is vaguely the right place to draw the line.

Wait when:

An example is incorrect

A table has some wrong entries

A technical assumption is missing in the statement of a lemma, but everywhere the lemma is used the assumption holds

You say something's you've constructed is unique, but really it's only unique up to complex conjugation.

You're proving some result subject to several technical assumptions, turns out you forgot to include a technical assumption that's no worse than the others.

Contact immediately when:

A result (not just a lemma) in the paper is completely wrong

A result in the paper may be correct, but the proof is unfixable

A section needs to be removed or added

A construction you use doesn't work, it needs to be replaced by a rather different construction.

If the error is trivial and you think the referee will overlook it (missing factor of 2, etc), you could wait. If the error is obvious and you think the referee will see it at once, (you forgot to say the manifold is assumed to be compact, and the theorem obviously fails for $\mathbb{R}^n$), you could wait. Otherwise, as a referee, I'd rather know about it. I tend to read papers very carefully and sometimes spend a lot of time deciding whether some obscurity is due to an error or my own ignorance (I'm a younger mathematician and will probably outgrow this eventually :) A constant stream of corrections would be annoying, but one significant one can be appreciated.

If the error will require major revisions or seriously weaken the result, I'd say notify for sure. If you aren't yet sure of the fix and think you might need to withdraw the paper, notify for sure, so the referee needn't waste time reading it in the meantime.

I somewhat disagree with the answers given so far. There are great differences among mathematicians (at the cultural level, and also at the individual level) in their attitude to errors in written work. Some people -- dare I mention continental Europeans? -- are really distressed by coming across a false statement in a paper. It makes them think that the author is at best "sloppy"; worse, there is the risk that an error that looks silly to you may cause the referee to lose confidence in your work.

Also remember that you know what you meant 100 times more than anyone else, and someone who is struggling to understand an argument which is new to them may not be so willing to correct a trivial error. In the intermediate level graduate course I am teaching now I am writing up all of my own notes and my own exercises. Several students in my class write to me periodically to point out trivial errors in my notes: e.g. missing "not"s and such things. Indeed, I encourage them to do so, because I know that these errors still give my students some trouble: their guesses at what I meant to say are usually correct, but they lack conviction and are thus somewhat nervous that they are misinterpreting something.

So I would say that if you discover that you have made an error in your paper -- again, I mean an incorrect mathematical statement, not a missing citation or a piece of exposition that you have found is more obscure than you wanted -- then you should contact the editor promptly. If it is easy to correct the error, do so, and enclose the new version of the paper and/or link to your homepage. The editor and the referee can decide what is truly minor.

Of course, when you contact the editor to say that you have corrected something and made a new version, it's a great time to recheck the rest of the paper. Sending one or two such "author generated revisions" on a given paper is (in my opinion, of course) being a diligent author. More than that really does risk annoying and eroding the confidence of the editor and referee.

Addendum: I see from the comments that Kevin Buzzard has expressed the opposite opinion, and he -- unlike me -- is an editor (although I have been a referee many times over and do not mind receiving an unsolicited revision). So now I'm thinking that the correct answer may depend upon the editor, and that there is no one best policy.

In reply to the addendum: Would you suggest asking the particular editor directly, then, whether s/he would object to forwarding a revision?
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Mike ShulmanApr 21 '10 at 18:56

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No, not exactly. I would guess that Kevin does not object to forwarding revisions; he just finds them annoying. Here's an idea: along with the new revision, you also send a short text file briefly explaining the minor errors that you caught. Then you tell the editor that a new revision addressing these errors is available at www.whatever, and you leave it up to him/her whether to forward that on to the referee.
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Pete L. ClarkApr 21 '10 at 19:24

This is a slightly silly comment, but I'll explain why I think it's worth making.

The best answer, of course, is don't make mistakes!

Of course, we're all human (and, ahem, it's not like I haven't had to correct papers which have already been published). However, I continue to be amazed by how sloppy some papers are which come to me as a referee. These are often by serious, established mathematicians. But I'm talking about people managing to cite their own work incorrectly (or get a definition, which they invented, wrong), or major, paper-breaking errors which I, as the referee, spot almost immediately. I can only imagine that people, once they have proved a result, almost fall over themselves to write it up and send it off. I'm young, and relatively patient, but I imagine that this behaviour is a good way to get rejected out of hand (even if the, corrected, paper is quite nice).

To answer the original question: as a referee, I would prefer corrections, but I would agree with Pete Clark that this would make me lose faith in the author(s). If I were an editor: I don't know...

"we're all human": If you speak of authors of mathematical paper then, technically, this is not true (math.rutgers.edu/~zeilberg/ekhad.html). Speaking of, e.g., MO users I am not sure either…
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DirkMay 2 '14 at 6:49

I second the answers that have already been given, but you might also consider updating the paper wherever it appears on the internet (your website, the arxiv). The referee(s) might not have started looking at your paper yet, and when they do, there's a decent chance that they'll download the paper from the arxiv or your website rather than looking at the copy that was actually sent to them by the journal.

(This comes from experience. I once had a paper where the referee ended up reading a version that wasn't quite final. I had caught some typos between putting the paper on the arxiv and sending to the journal; but the referee read the arxiv version, so the typos got caught again. This I think one can live with -- I wouldn't update an arxiv submission just to fix typos until the final version -- but if the changes were serious I think it would have been worthwhile.)

It's really surprising to me that a referee would do that. When I am refereeing a paper I always make sure to read the version sent to me by the journal, for precisely the reason you mention -- the author may have made corrections but not bothered updating the arxiv version until it was final.
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Mike ShulmanApr 21 '10 at 18:47

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I'm also surprised. I would never dream of refereeing anything other than the version sent to me by the journal! How do you expect the editors to deal with a report that says "there's an error in section 7" if the submitted version only has 6 sections?? This has the potential to go disastrously wrong!
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Kevin BuzzardApr 21 '10 at 20:35

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You think that's disastrous? I had a referee reject a completely different paper of mine! I assume they must have downloaded a random paper of mine off the arxiv. They wrote quite a scathing rejection of the paper without being very specific - fortunately I noticed that they alluded to certain things that were not mentioned in the submitted paper so I was able to point this out to the editor. (And it all ended happily for both papers.)
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Simon WillertonApr 22 '10 at 1:28

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I, too, had a referee reject the wrong paper -- one that had already been published!
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Allen KnutsonApr 25 '10 at 16:13

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One of my early refereeing mistakes was to do exactly this: after I finished writing my report on the paper with the same title I downloaded from the author's site, I took a look at the paper I had been emailed, to note that this paper was completely different: I had to largely rewrite my report, and changed my recommendation from reject to accept. Never again!
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Charles StewartJun 4 '10 at 11:58